Vol. 3, No. 1, 2020

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Poincaré Duality in Dimension 3

Jonathan A. Hillman


Poincaré duality is central to the understanding of manifold topology. Dimension 3 is critical in various respects, being between the known territory of surfaces and the wilderness manifest in dimensions 4. The main thrust of 3-manifold topology for the past half century has been to show that aspherical closed 3-manifolds are determined by their fundamental groups. Relatively little attention has been given to the question of which groups arise. This book is the first comprehensive account of what is known about PD3-complexes, which model the homotopy types of closed 3-manifolds, and PD3-groups, which correspond to aspherical 3-manifolds. In the first half we show that every P2-irreducible PD3-complex is a connected sum of indecomposables, which are either aspherical or have virtually free fundamental group, and largely determine the latter class. The picture is much less complete in the aspherical case. We sketch several possible aproaches for tackling the central question, whether every PD3-group is a 3-manifold group, and then explore properties of subgroups of PD3-groups, unifying many results of 3-manifold topology. We conclude with an appendix listing over 60 questions. Our general approach is to prove most assertions which are specifically about Poincaré duality in dimension 3, but otherwise to cite standard references for the major supporting results.

Target readership: graduate students and mathematicians with an interest in low-dimensional topology.